As for the attractor indices
which enter these formulae,
it was conjectured in [1]},
that for quivers \((Q,W)\)  associated to a non-compact CY threefolds of the form \(\mathfrak {Y}=K_{S}\)  (namely, the total space of the canonical bundle over a Fano surface \(S\) ),
the attractor invariants \(\Omega _\star (\gamma )\)  take a very simple form:
\(\Omega _\star (\gamma )=0\)  except when \(\gamma \)  corresponds to a dimension vector supported on one node of the quiver (in which case \(\Omega _\star (\gamma )=1\) ), or \(\gamma =k\delta \)  with \(k\ge 1\)  and \(\delta \)  the charge vector for the skyscraper sheaf (in which case \(\Omega _\star (k\delta )=-\chi _{\mathfrak {Y}}\) , the Euler number of \(\mathfrak {Y}\) ). This Attractor Conjecture (AC) was arrived at by comparing the quiver indices with
the counting of Gieseker-semi-stable sheaves on \(S\) , and supported by an analysis of the expected
dimension of the moduli space of quiver representations in the self-stability chamber.
Further evidence and an extension of AC to all toric CY three-folds was presented in [2]}, [3]}. In this work, we focus on the simplest case \(\mathfrak {Y}=K_{{P}^2}\)  (also known as local \({P}^2\) ), which is a crepant resolution of the orbifold singularity \({C}^3/{Z}_3\) 
and whose derived category of (compactly supported) coherent sheaves is isomorphic to the
derived category of a three-node quiver \((Q,W)\)  shown in Figure REF .
Combining ideas from [1]}, [5]}, we prove
in §REF  that the Attractor Conjecture holds for this quiver, thereby
providing the attractor invariants relevant in the vicinity of the orbifold point.
